26  The complex exponential

The functional equation for the exponential and the Cauchy–Riemann equations determine the complex exponential.

In this section, we give some relatively easy exercises to illustrate the important concepts of complex analytic functions.

In this exercise, we derive the complex exponential function. The starting point is the real exponential function, which is the unique function \(f(x) = \exp(x)\) that satisfies \[f(x_1 + x_2) = f(x_1)f(x_2), \quad f(0) = 1.\] We desire to generalize \(\exp(x)\) to the complex plane, so that \(\exp(x + 0\mathrm{i}) = \exp(x) \in \mathbb{R}\).

To avoid confusion, we call this unknown function \(f : \mathbb{C}\to \mathbb{C}\).

Exercises

Exercise 26.1 Show that \(f(x + \mathrm{i}y) = \exp(x) f(\mathrm{i}y)\). (Thus, we need to determine \(f(\mathrm{i}y)\).)

Exercise 26.2 We set \(f(\mathrm{i}y) = A(y) + \mathrm{i}B(y)\). Show that \[A(y) = B'(y), \quad B(y) = - A'(y),\] and hence that \(A''(y) = -A(y)\). Hint: Cauchy–Riemann.

Exercise 26.3 The general solution to the ODE \(A'' = -A\) is \[A(y) = \alpha \cos(y) + \beta\sin(y),\] where \(\alpha\) and \(\beta\) are constants to be determined. Show that \(\alpha = 1\) and \(\beta = 0\) are the only constants compatible with \(f(z)\) being a generalization of the real exponential function.

Exercise 26.4 Conclude that \[f(x + \mathrm{i}y) = e^x(\cos y + \mathrm{i}\sin y)\]

Exercise 26.5 Show that \((\exp(z))' = \exp(z)\), using Cauchy–Riemann, and that \(\exp(z)\) is everywhere analytic.

Exercise 26.6 Show that the equation \[\exp(z) = w\] has infinitely many solutions for any \(w\neq 0\).

Exercise 26.7 Show that \(\exp(z) \neq 0\).